Séparation des représentations par des surgroupes quadratiques

Abstract

Let $\pi$ be an unitary irreducible representation of a Lie group $G$. $\pi$ defines a moment set $I_{\pi }$, subset of the dual ${\mathfrak{g}}^{\ast }$ of the Lie algebra of $G$. Unfortunately, $I_{\pi }$ does not characterize $\pi$. However, we sometimes can find an overgroup $G^{+}$ for $G$, and associate, to $\pi$, a representation ${\pi }^{+}$ of $G^{+}$ in such a manner that $I_{{\pi }^{+}}$ characterizes $\pi$, at least for generic representations $\pi$. If this construction is based on polynomial functions with degree at most 2, we say that $G^{+}$ is a quadratic overgroup for $G$. In this paper, we prove the existence of such a quadratic overgroup for many different classes of $G$.